We will once again rely on the fact that \(x\perp y\)  if and only if \(y\in \mathop {\mathrm {Ker}}f_x\) , where \(f_x\)  is a
supporting functional at \(x\) .
(iii) \(\Longrightarrow \)  (ii). Choose an arbitrary \((n-1)\) -tuple of lines \((x_1,\dots ,x_{n-1})\) . Let \(f_i\)  be any supporting functional at \(x_i\) , \(i=1,\dots ,n-1\) . Denote \(X_{n-1}=\bigcap _{i=1}^{n-1}\mathop {\mathrm {Ker}}f_i\) .
Then \(\mathrm {codim}\,X_{n-1}\le n-1\)  together with \(\dim X\ge n\)  imply that the subspace \(X_{n-1}\subseteq X\)  contains at least one line, \(x_n\) . Clearly, \(x_i \perp x_n\)  for \(i = 1,\dots ,n-1\) .
(ii) \(\Longrightarrow \)  (i) is straightforward.
\(\lnot \)  (iii) \(\Longrightarrow \)  \(\lnot \)  (i). Assume \(\dim X=k\le n-1\) . From [1]} and the fact that the norm is positively homogeneous and \(x\mapsto \lambda x\)  is a differentiable map it follows that the norm's unit ball has a
smooth point \(\hat{x}_1\) . Let \(f_1\)  be a unique supporting functional at \(\hat{x}_1\)  and let
\(x_1:=[\hat{x}_1]\) 
be the corresponding line in \(\hat{\Gamma }(X)\) . Then \(x_1\perp y\)  for a line \(y\)  is equivalent to \(y\subseteq \mathop {\mathrm {Ker}}f_1=X_1\) .
Now, the induced norm in the \((k-1)\) -dimensional space \(X_1\)  again has a smooth point, say \(\hat{x}_2\) ;
let \(f_2\colon X_1\rightarrow \mathbb {F}\)  be the corresponding supporting functional which by the Hahn–Banach theorem we
extend to a supporting
functional at a line \(x_2:=[\hat{x}_2]\)  on the original space \(X_0:=X\) . Recursively, one obtains a \(k\) -tuple of lines
\((x_1,\dots ,x_{k})\) , where \(x_i\subseteq X_{i-1}\subseteq X_{i-2},\)  \(2\le i\le k,\)  is
a line spanned by a smooth point \(\hat{x}_i \in X_{i-1}:= X_{i-2}\cap \mathop {\mathrm {Ker}}f_{i-1}\)  and has a unique supporting functional \(f_i\colon X_{i-1}\rightarrow \mathbb {F}\)  which we again extend by the Hahn–Banach theorem to a supporting functional defined on \(X\) . Clearly, \(i<j\) 
implies that \(x_j\subseteq \mathop {\mathrm {Ker}}f_i\) , so \(x_i\perp x_j\)  for \(1\le i<j\le k\) . Assume a line \(x\subseteq X\)  satisfies \(x_i\perp x\)  for each \(1\le i\le k\) . Since \(x_1\perp x\)  and
\(\hat{x}_1\in X\)  is smooth, then \(x\subseteq \mathop {\mathrm {Ker}}f_1=X_1\) . Since \(\hat{x}_2\in X_1\)  is smooth and \(x_2\perp x\) , we
further have \(x\subseteq \mathop {\mathrm {Ker}}f_2\cap X_1=X_2\) . Proceeding recursively, one finds
\(x\subseteq X_k=0\) , a contradiction.
